Chapter 3 arXiv:2010.16239v1 [cond-mat.mtrl-sci] 29 Oct 2020 · for the boron atoms to describe the dynamics of the crystal lattice. 3.1 Introduction There is a wide variety of borides

November 2, 2020 1:45 PSP Book - 9in x 6in Book_Borides

Chapter 3

Crystal structures of dodecaborides:complexity in simplicity

Nadezhda B. Bolotinaa,∗, Alexander P. Dudkaa,b,Olga N. Khrykinaa,b, and Vladimir S. Mironova

aShubnikov Institute of Crystallography of the Federal Scientific Research Center“Crystallography and Photonics” of the Russian Academy of Sciences, Leninsky Prospekt59, 119333 Moscow, RussiabProkhorov General Physics Institute, Russian Academy of Sciences, Vavilova str. 38,119991 Moscow, Russia∗E-mail address: [email protected]

Abstract

Analysis of the intriguing physical properties of the dodecaborides,RB12, requires accurate data on their crystal structure. We showthat a simple cubic model fits well with the atomic positions in theunit cell but cannot explain the observed anisotropy in the physicalproperties. The cooperative Jahn-Teller (JT) effect slightly violatesthe ideal metric of the cubic lattice and the symmetry of the electrondensity distribution in the lattice interstices. Theoretical modelsof the JT distortions of the boron framework are presented. Theircorrespondence to the electron-density distribution on the maps

Rare-Earth BoridesDmytro S. Inosov (ed.)Copyright © 2020 by Jenny Stanford Publishing Pte. Ltd.www.jennystanford.com

arX

iv:2

010.

1623

9v1

[co

nd-m

at.m

trl-

sci]

29

Oct

202

0

mailto:[email protected]


2 Chapter 3 Crystal structures of dodecaborides: complexity in simplicity

of Fourier syntheses obtained using x-ray data and explainingthe previously observed anisotropy of conductive properties isdemonstrated. The effect of boron isotope composition on thecharacter of the lattice distortions is shown. We also discuss theapplication of the Einstein model for cations and the Debye modelfor the boron atoms to describe the dynamics of the crystal lattice.

3.1 Introduction

There is a wide variety of borides formed by different-in-shapeboron polyhedra in combination with most metals, resulting in ametal-to-boron ratio ranging from 4:1 to 1:66. A highly symmetricalFm3m structure of the uranium dodecaboride UB12 was the firstone of this type [1]. A unit cell of the cubic lattice with the latticeconstant a = 7.473 Å contains four UB12 formula units. The metalatoms occupy centers of truncated B24 octahedra with boron atomsat each of their 24 vertices. Every boron atom is bonded to twometal atoms and to five other boron atoms. The metal and boronatoms are in positions 4a {0, 0, 0} and 48i {0.5, y, y} of the Fm3mgroup, respectively, with y close to 1/6. This compound may also

Figure 3.1 Left: two metal-centered truncated B24 octahedra of the UB12-type structure, connected by one empty B12 cuboctahedron. Right: Thesame structure presented as a NaCl-type structure. The metal atoms (largespheres) alternate in a checkerboard pattern with the B12 cuboctahedra.


3.1 Introduction 3

be described in terms of the NaCl-type structure, in which metalatoms and regular B12 cuboctahedra occupy the Na and Cl posi-tions, respectively, resulting in a face-centered cubic (fcc) structureshown in Fig. 3.1.

Three years later, the isomorphous ZrB12 was prepared andstudied [2]. Six UB12-type structures of the rare-earth dode-caborides RB12 (R = Y, Dy, Ho, Er, Tm, Lu) were determined [3]based on the x-ray powder diffraction data. The authors of Ref. [4]reported the cubic symmetry after examination of a powder ScB12sample. The cubic structure of YB12 was later confirmed on sin-gle crystals [5], but a single crystal of ScB12 studied in the samework [5] was determined as tetragonal I4/mmm, with unit-cellparameters atet ≈ 5.22 Å, ctet ≈ 7.35 Å that could be transformedto pseudo-cubic: a = b = atet

√2 ≈ 7.38 Å, c = ctet ≈ 7.35 Å.

All the above findings were summarized in a review article [6]. Amore recent review [7], which was mainly devoted to magnetic,superconducting, and other physical properties of rare-earth do-decaborides, began by describing the dodecaboride structure andby presenting structural information on RB12 with R = Tb – Lufrom the second half of the lanthanide series supplemented withYB12 and ZrB12. Background information on higher borides ofrare earths, including dodecaborides, has been summarized in [8].It is worth noting that the rare-earth dodecaborides RB12 differfrom other higher borides RBn, n > 12. All the dodecaborides, ex-cept YbB12, are good metals similar to RB6 and RB4, whereas RBnwith n > 12 are insulators. Both B12 cuboctahedra and icosahedraare electron-deficient by two electrons. The trivalent rare-earthatoms can supply three electrons, so there is one excess conduc-tion electron per unit cell. The only exception is the narrow-gapsemiconductor YbB12, known also as a Kondo insulator, where Ybtakes an intermediate valence. The summary table of structural,electronic and magnetic characteristics of the dodecaborides of theUB12 type different in isotope boron composition is presented inRef. [9]. In addition to the rare-earth dodecaborides TbB12 – LuB12mentioned above, this table contains ZrB12, HfB12, pseudo-cubicScB12, GdB12 synthesized under high pressure [10] as well as dode-caborides of heavy metals ThB12, UB12, NpB12, PuB12 and severalsolid solutions R1xR21−xB12 of the UB12 type. In the same arti-



cle [9], the lattice constants of RB12 as well as the B-B distances inand between the B12 cuboctahedra versus ionic radii of the metalatoms are plotted and discussed. A large amount of the referencedata on the structures and properties of higher borides, includingdodecaborides RB12, is contained in the overview chapter of arecent PhD thesis [11]. Metal dodecaborides MB12 attract particu-lar interest as multifunctional materials. For instance, in contrastto conventional superhard materials like diamond, which are in-sulators or semiconductors, many dodecaborides are superhardcompounds with high electrical conductivities that can be used asconductors at extreme conditions [12].

Owing to the simple cubic structure, dodecaborides are conve-nient objects for studying physical properties of the metal atomspossessing relative freedom in the large B24 cavities of the boronframework. There is a large number of publications on the physi-cal properties of dodecaborides. Their crystal structures, however,have not been studied in such detail and are almost not stud-ied at low temperatures (the ZrB12 structure at 140 K [13] is arare exception), although low-temperature physical properties of-ten reveal features that require explanations based on the crystalstructure. Moreover, clear explanations are not always easy to ob-tain in the framework of a simple cubic model. For example, inRefs. [14, 15], linear thermal expansion coefficients α of RB12 singlecrystals, R = Y, Ho, Er, Tm, Lu, were measured in the temperaturerange of 5–300 K. The values of the coefficients at low temperaturesvaried nonlinearly for all the compounds studied. The nonlinearα(T) dependencies had two minima, sometimes negative in magni-tude. Two temperature intervals with negative thermal expansion(NTE) were found for LuB12 crystals: the first one was 60–130 Kwith a minimal negative at 90 K and the second one was 10–20 Kwith a minimum at 12 K. One NTE interval 50–70 K was revealedfor YB12 with a negative minimum near 60 K whereas the secondminimum at 15 K was close to zero but positive.

The relationship between anomalies of the thermal expansionand crystal structure of the dodecaborides is poorly understood sofar. Until recently, published data on the observed anomalies of theRB12 structure was actually limited to discussions about observeddisordering of the metal atoms near the 4a position of the Fm3m


3.2 Introduction 5

group [16, 17] and to a short report on a small tetragonal distor-tion of the LuB12 lattice at temperatures below 150 K [18], whichwas discovered in the analysis of thermal expansion using x-raydata. Reports of a tetragonal distortion of the dodecaboride latticeeven at room temperature appeared before, but they only con-cerned ScB12 and Sc-containing solid solutions Sc1−xRxB12 (R = Y,Zr) [5, 12, 19, 20]. As noted in [20], the origin of the transforma-tion from cubic to tetragonal structure in ScB12 is unclear. Theinfluence of the size of scandium is not obvious, as the radius ofscandium is located within the limits of atomic radii of metalsthat form cubic RB12 phases. The cell parameters of known dode-caborides are graphically presented in [20] as functions of theird, 4 f , or 5 f metal radii. All data fit on three straight lines withan individual slope for each group of elements. Yttrium, gadolin-ium and thorium dodecaborides that finish the correspondingseries have almost identical radii but considerably different lat-tice constants. The d-elements form the Hf – Zr – Sc – Y line withintermediate Sc which enters into the composition of the tetrago-nal ScB12 structure. Structural stability and physical properties ofMB12 containing transition-metal elements (M = Sc, Y, Zr, Hf) werestudied [12] using first-principles calculations supplemented withthe x-ray diffraction experiments for ScB12 and YB12. The tetrag-onal I4/mmm structure was predicted to be the thermodynamicground state of ScB12 and a metastable state of YB12, ZrB12, andHfB12. Tetragonal ScB12 was shown to transform reversibly intothe cubic Fm3m symmetry group at 510 K, which correspondedto the thermodynamic ground state of YB12, ZrB12, and HfB12 atroom temperature. The temperatures of the phase transition be-tween tetragonal and cubic phases of the yttrium and zirconiumdodecaborides could be lower than 100 K as predicted in [12].

Neutron diffraction measurements have revealed that HoB12,TmB12 and ErB12 have incommensurate magnetic structures[21–23]. The complex magnetic structure of these materials seemsto result from the interplay between the RKKY and dipole-dipoleinteractions. Strong frustration of an antiferromagnetic order inthe fcc symmetry of the dodecaborides could also play an impor-tant role.



3.2 Cooperative Jahn-Teller effect as a driving forcebehind structural instability in dodecaborides

The origin of small structural distortions in RB12 dodecaborides,which distinctly manifest in the background of an almost un-changeable basic cubic structure of the robust boron network, isa challenging and intriguing problem. In most of the works onrare-earth dodecaborides RB12, discussion of structural instabilityis usually based on the two main points: (a) the boron networkis rigid and undistorted, (b) the dimension of the B24 cavity isoversized with respect to the central metal ion R; this results ina rattling character of thermal vibrations of the metal atoms R inthe cavities [9, 17, 24–26]. According to this approach, at low tem-peratures some fraction of the metal ions shifts from the centralposition in the B24 cages to form a cage-glass state [24–27]. Thesedisplacements may be caused by lattice defects such as impuritiesand boron vacancies. These models were extensively used in theanalysis of low-temperature structural and magnetic characteris-tics of RB12 compounds [24–27].

However, as will be shown below, the basic assumptions ofthese models need some revision, since they do not take intoaccount some important features of the electronic structure ofthe RB12 dodecaborides and their structural units. Namely, apartfrom displacements of R atoms in the oversized B24 cages, low-temperature lattice distortions in RB12 may originate from intrinsicstructural instability of B12 cuboctahedra related to the JT effect.This point can be best illustrated for isolated B12 units. Similarly toother high-symmetry molecules, such as B12 icosahedra in boronand higher metal borides [28, 29], the cuboctahedral boron clustersB12 may have an orbitally degenerate ground state resulting in JTdistortions of the regular cubic structure. In terms of molecularorbitals (MOs), the orbital degeneracy of the ground state is as-sociated with partial electron occupation of the highest occupiedmolecular orbital (HOMO) of B12, which is represented by triplydegenerate MOs of a t-type symmetry (Fig. 3.2).

Essentially, the character of the JT distortion depends on theelectric charge of the B12 clusters. In an electrically neutral [B12]0


3.2 Cooperative Jahn-Teller effect as a driving force behind structural instability in . . . 7

cluster, the HOMO accommodates two electrons (t2 configuration);in negatively charged clusters [B12]n− (n = 1–4), the number m oft-electrons in HOMO increases from three to six (Fig. 3.3). Fourelectronic configurations tm with m = 2–5 produce a triply degen-erate many-electron ground T-state [Fig. 3.3 (a–d)] This impliesthat neutral [B12]0 cluster and charged [B12]n− clusters (n = 1, 2, 3)are JT-active systems, which would tend to distort the cubic struc-ture, except the [B12]4− cluster with fully occupied HOMO (t6

configuration), which has a nondegenerate ground state and thusis not JT-active [Fig. 3.3 (e)]. For JT systems with the T-type groundstate, the character of distortions is determined by two active vi-brational modes of e and t2 symmetry; such situation is referred toas the T× (e + t2) Jahn-Teller problem [30]. In this case, depending

o Tetragonal JT minima (D4h)

o Trigonal JT minima (D3d)

o Saddle points

JT problem

Electronic

Т-termJT-active vibrational modes

of e- and t- symmetry

Potential energy surface of

Т-type JT system involves:

The highest occupied molecular orbital (HOMO)

of B12 cluster is triply degenerate

Cuboctahedral boron cluster B12 is a triply degenerate

Jahn-Teller system (JT T-system)

Figure 3.2 On the origin of the Jahn-Teller effect in isolated cuboctahe-dral B12 clusters in RB12 compounds. The highest occupied molecularorbital (HOMO) is triply degenerate. Partial population of the HOMOorbitals with electrons produces a triply degenerate many-electron groundT-state, which leads to Jahn-Teller distortions.



Figure 3.3 Molecular structure of neutral and negatively chargedisolated [B12]n− clusters (n = 0–4) obtained from DFT geometry-optimization calculations [27]. The optimized structures correspond to thedeeper local JT minima of the corresponding charged cluster. Principalatomic distances (Å) and bond angles are indicated.

on the ratio between the strength of the e and t2 electron-vibroniccouplings, three types of minimum points on the ground-statepotential energy surface can occur: trigonal (D3d), tetragonal (D4h)and orthorhombic (D2h) points (Fig. 3.2) [30].

More quantitative information on the amplitude and type ofthe JT distortions has been obtained from density functional the-ory (DFT) calculations for the neutral cluster [B12]0 and negativelycharged [B12]n− clusters (n = 1–4) [27]. Calculated structures of[B12]n− (n = 0–4) clusters resulting from the DFT geometry op-timization are shown in Fig. 3.3. These results indicate that theJT-active clusters [B12]n− (n = 0–3) are slightly distorted cuboc-tahedra. The character of JT distortions depends strongly on thecharge of the cluster: the neutral cluster and charged clusters withn = 1, 2 exhibit trigonal type of JT distortions, while the [B12]3−

cluster shows tetragonal JT distortion. It is important to note thatthe overall magnitude of the JT distortions in isolated B12 clustersis rather small, as the bond lengths and bonding angles vary within



∼0.1 Å and∼5◦, respectively (Fig. 3.3); DFT calculations show thatthe energy gain resulting from JT distortions is within 0.2–0.3 eVper B12 cluster.

These results suggest that the JT structural lability of the B12units in the crystal lattice of metal dodecaborides RB12 may play animportant role in the microscopic mechanism of lattice distortionsof RB12 at low temperature. It should be borne in mind, however,that in the actual crystal structure of RB12, the B12 clusters areconnected by B-B covalent bonds to form an extended 3D covalentboron network, in which HOMOs of individual B12 clusters mayhave non-integer electron occupation. Nevertheless, one can expectthat in RB12 crystals some fraction of the JT activity of B12 clus-ters may retain in the three-dimensional boron network becausethe local triply degenerate HOMOs of B12 cuboctahedra remainpartially filled, as can be seen from the overall electron balancebetween the metal and boron sublattices. Due to interactions be-tween the nearest B12 clusters in a RB12 crystal, local JT distortionsof B12 cuboctahedra become mutually consistent resulting in asymmetry-lowering distortion of the lattice; this phenomenon isknown as the cooperative JT effect, which is well documented inthe literature [30–32].

In a concentrated JT system, the full JT Hamiltonian of thecrystal is given by the equation [31, 32]:

H = ∑n

HJT(n) +12 ∑

n,m(n 6=m)

Q+(n)K(n−m)Q(m), (3.1)

where the vector indices n and m enumerate unit cells of the crys-tal, HJT(n) is the one-center JT Hamiltonian for unit cell n, andthe last term represents pairwise interactions between the localJT centers n and m. Here Q(n) is a vector whose components arethe local JT-active vibrational modes and K(n−m) is the opera-tor describing interactions between the local JT vibrational modeson sites n and m. The electronic and geometric structure of a co-operative JT system is determined from the minimization of thetotal energy of the crystal resulting from the competition of thelocal distortions, the first term in Eq. (3.1), and the interactionbetween the different sites (the second term). In the general case,



cooperative JT interactions can lead to a variety of situations, de-pending on the specific character of orbitally-degenerate moietiesand interplay between the on-site and inter-site JT interactions. Inparticular, lower symmetry structures in RB12 resulting from thesecooperative interactions can lead to a parallel alignment of all thelocal distortions of B12 cubooctahedra (which is termed as ferrodis-tortive case) or to a more complicated geometrical arrangement ofthe local B12 distortions (so-called antiferrodistortive case). In theferrodistortive phase, the local JT centers are coupled to a strainof the lattice, which changes the shape of the crystal and its unitcell parameters; this strain mode coupling provides an effectivelong-range interaction between the JT centers.

It is important to note that in most cooperative JT systems thecoupling is predominantly to a strain of the lattice [31, 32]. Thisfact gives an idea of the origin of observed anomalous behavior ofRB12 dodecaborides: in fact, formation of a ferrodistortive JT phasewith a long-range ordering of JT distortions mediated by the strainof the lattice is the most likely scenario in metal dodecaborides.

Figure 3.4 Structure of the ferrodistortive JT phase in RB12 dodeca-borides and the character of the strain of the crystal lattice. (a) The B12cuboctahedra are all elongated along the trigonal axis [111] and com-pressed in the orthogonal plane; this geometry corresponds to one of thelocal trigonal JT minima of B12 shown in Fig. 3.3; (b) The crystal lattice ofRB12 is compressed in the (111) plane.



More specifically, the ferrodistortive JT phase of RB12 with theelastic strain axis parallel to the [111] direction seems to be theactual situation that may account for the unusual behavior of RB12.In this case, when the strain mode corresponds to elongation, theB12 cuboctahedra are all elongated along the trigonal axis [111],that refers to one of the trigonal JT minima shown in Fig. 3.3; thelocal distortion JT axes are all parallel to each other [Fig. 3.4 (a)].Since the elastic strain does not change the volume of the crystal,elongation in the [111] directions is followed by compression inthe (111) plane, as depicted in Fig. 3.4 (b).

This leads to important changes in the electronic band structurecaused by the trigonal strain mode, as the compression in the (111)plane gives rise to some shortening in the R-R distance betweenthe neighboring metal atoms [Fig. 3.5 (a,b)]. This would changeorbital interactions between the R and B atoms responsible forthe formation of the electronic conduction band of RB12, whichis mainly represented by 2p(B) and 5d(R) atomic orbitals. Thelargest changes are expected for the 5dz2 metal orbitals that havethe strongest σ-type overlap with the 2p valence orbitals of boron[Fig. 3.5 (c)]. Accordingly, enhanced 5dz2(R)-2p(B) orbital overlapincreases the energy dispersion of the electronic conduction band,thereby increasing the overall number of the filled conduction bandstate below the Fermi level of RB12. This results in larger electronpopulation of the 5dz2 (R) orbitals oriented along the local R-R linesconnecting neighboring R atoms in the (111) plane [Fig. 3.5 (c)].Considering the elongated shape of 5dz2(R) orbitals, this leads toincreased electron density along the R-R lines being parallel tothe side diagonals [110], [011], [101]), which are shown with redsolid lines in Fig. 3.6. These findings are in excellent agreementwith the recent experimental results on LuB12, which reveal lower-symmetry electron density distribution (charge stripes) correlatingwith the filamentary structure of conduction channels observed inthe magnetoresistance measurements [33]. Remarkably, the generalcharacter of the residual density distribution near the metal atomin the (100) and (010) planes at low temperatures (50 K) shown inFig. 3.15 strongly resembles the shape of 5dz2(R) orbitals orientedalong the R-R lines, as depicted in Fig. 3.5 and Fig. 3.6.



Figure 3.5 On the origin of charge stripes in the ferrodistortive JT phasein RB12 dodecaborides, (a) The general character of the local distortions inthe ferrodistortive JT phase, (b) reduction of the R-R distance between theneighboring metal atoms due to compression in the (111) plane, (c) ori-entation of 5dz2 metal orbitals having the strongest σ-type overlap withthe 2p valence orbitals of boron; shortening of the R-R distance leads toa maximal change in the 5dz2 (R)-2p(B) orbital overlap, which ultimatelycauses transfer of excess electron density to the 5dz2 orbitals and formationof the charge stripes (see Figs. 3.15 and 3.16 later in the text).

Thus, a theoretical model based on the cooperative JT effectprovides a new insight into the microscopic origin of the myste-rious structural behavior of RB12 cubic dodecaborides. The mainreason behind the manifestation of low-symmetry effects of RB12lies in the inherent structural lability of the B12 cuboctahedral units



Figure 3.6 Orientation of the charge stripes (the solid red lines) in the(111) plane of RB12 resulting from the electron density transfer to the5dz2 (R) orbitals caused by the elastic strain of the crystal lattice in theferrodistortive JT phase (see Fig. 3.5).

resulting from their orbital degeneracy and the JT effect. Assuminga ferrodistortive JT ordering in RB12 with deeper trigonal JT min-imum of B12 clusters enables one to rationalize in a natural waythe main experimental results on dodecaborides (see followingsections of this chapter for more details):

(1) The cooperative JT model explains the presence of small distor-tions of the cubic lattice of RB12 at all temperatures, includingroom temperature. Indeed, in the ferrodistortive JT state dis-tortions persist at all temperatures, without a structural phasetransition. Theoretical treatment of the ferrodistortive JT stateis similar to that for a magnetic spin lattice in an external mag-netic field, which always retains some magnetization [31, 32];the latter serves as an order parameter, whose analog in RB12dodecaborides corresponds to the relative magnitude of thedeviation from the regular cubic structure.



(2) The JT model provides a physically transparent insight intothe origin of lower symmetry electron density distributionin RB12, including appearance of charge stripes in LuB12 andthe filamentary structure of conduction channels resulting inanisotropic magnetoresistance [33]. These effects are mainlydue to enhanced electron occupation of the 5dz2 metal orbitalsresulting from the larger 5d(R)-2p(B) orbital overlap causedby elastic shortening in the (111) plane in the ferrodistortive JTstate (Figs. 3.4 and 3.5).

(3) The above consideration indicates that low-symmetry distor-tions are a unique property of all rare-earth dodecaborides, asthey result from the JT structural lability of B12 units, not fromthe ground-state characteristics of the metal ions R.

(4) This theoretical approach evidently demonstrates that subtlestructural departures from the regular cubic structure of RB12dodecaborides are by no means an artifact of x-ray diffractionanalysis, but they are an inherent property of all rare-earthRB12 compounds resulting from the JT activity of the B12 units.Indeed, in the ferrodistortive JT state, arbitrarily weak JT inter-actions always lead to a static lattice distortion, by analogy withthe nonzero magnetization of the spin system in an externalmagnetic field. More specifically, this property originates fromthe fact that the energy gain resulting from the JT distortionsgrows linearly with the strain value e, while the competingelastic energy is proportional to e2; therefore, at small strainsthe low-symmetry JT state lies lower in energy [30–32].

(5) In fact, the presence of significant 10B/11B isotope effects initself suggests the JT origin of the structural instability, since itdocuments a breakdown of the Born-Oppenheimer approxima-tion for the orbitally degenerate systems, in which electronicand vibrational motions are no longer independent. Generally,isotope effects are more pronounced for the dynamic JT effect,when the JT stabilization energy competes with the vibrationalenergy; this situation is likely to occur in RB12.


3.3 Modeling the dynamics of the dodecaboride lattice using x-ray diffraction data 15

3.3 Modeling the dynamics of the dodecaboridelattice using x-ray diffraction data

Experimental conditions, such as the sample temperature, can varyto better identify the barely perceptible symmetry violations, whichmay appear due to the cooperative JT effect. The most known phe-nomenon is the temperature dependence of the unit-cell values,which should be monotonous in the absence of a lattice transfor-mation (see below). Additional information can be obtained byanalyzing atomic displacement parameters (ADPs) at differenttemperatures using both experimental and theoretical temperaturecurves.

The key role in the structure analysis of crystals is assignedto structure factors F(H), which provide a transition from mea-sured intensities of diffraction peaks to the distribution of electrondensity in the crystal. The expression for F(H) is as follows:

F(H) =N

∑ν=1

fν(|H|) exp(2πirν ·H)Tν(H). (3.2)

Here H = ha∗ + kb∗ + lc∗ is a scattering vector; fν(|H|)is the atomic scattering factor of the atom at rν; Tν(H) =∫

p(uν) exp(2πiuν ·H)d3u is the temperature factor, which isknown also as the Debye-Waller factor that accounts for the atomicdisplacements uν from the lattice points. Summation is carried outover all atoms in the unit cell of the crystal. As one can see, the tem-perature factor Tν(H) is the Fourier transform of the probabilitydensity function p(uν) whose coefficients are atomic displacementparameters (ADPs) discussed below. The p(uν) function can beapproximated respectively by univariate or trivariate Gaussian incase of isotropic or anisotropic harmonic vibrations of an atom,and it can be more complicated in case of anharmonic vibrationsas a result of heating, for instance. In any case, however, the tem-perature does not participate directly in the calculations either asa fixed parameter or as a refined variable. Moreover, the conven-tional approach does not require any assumption of the atomicdisplacement nature. The ADP values may correspond to thermalvibrations supplemented with static shifts [34]. Along with theatomic coordinates, ADPs are the refined parameters of the struc-



tural model. The least-squares refinement procedure consists inapproximation of |Fcalc(H)|2 calculated by the formula (3.2) with|Fobs(H)|2, whose values are proportional to the measured intensi-ties of the diffraction reflections. The displacements of each atomare represented in the structural model by one or more parame-ters, depending on the chosen formalism (isotropic, anisotropicharmonic or anharmonic displacements). Harmonic ADPs forma second-rank matrix {uij}, 1 ≤ i, j ≤ 3, the trace of which givesan estimate of the equivalent atomic displacements 〈u2〉eq or ueqin short notation, ueq = (u11 + u22 + u33)/3. This parameter oftenappears in studies of the thermal properties of solids.

An alternative method of quantifying atomic displacement pa-rameters is not tied directly to a structural model. Thermal vi-bration amplitudes ucalc(R) of the metal atoms in the large cavi-ties of the dodecaboride structure well correspond to the Einsteinmodel [35] for independent harmonic oscillators supplementedwith a temperature independent static component 〈u2〉shift or ushiftin short notation:

ucalc(R) =h2

kBmaTE

(12+

1exp(TE/T − 1)

)+ ushift(R) (3.3)

The expanded Debye model [36] is suitable for atoms of the boronframework whose displacements strongly correlate with eachother:

ucalc(B) =3h2

kBmaTD

(14+

(T

TD

)2∫ TDT

0

y dyexp(y)− 1

)+ ushift(B)

(3.4)The agreed notations are: } = h/2π is the Planck constant; kB — theBoltzmann constant; ma — atomic mass; TE (TD) — the character-istic Einstein (Debye) temperature; T — the temperature of theexperiment.

The problem is that the characteristic Einstein (Debye) tem-perature and the value of ushift must be known in advance tocalculate the values of ucalc from Eqs. (3.3) or (3.4). Still, it is pos-sible to solve the inverse problem of calculating the characteristicEinstein (Debye) temperature and ushift using the values of ueqdetermined from diffraction data. For this purpose, one shouldcollect the multi-temperature data sets {h, k, l, |Fobs|, σF} and re-fine the crystal structure at each temperature using conventional



techniques. As a result, each atom is supplied at each temperaturewith a value of uobs = ueq. The multi-temperature set of theseparameters then serves as an input for a least-squares procedure∑ |u2

calc − u2obs| → min to fit the model curve to the set of uobs. The

characteristic Einstein (Debye) temperature TD (TE) and the valueof ushift are adjustable parameters of this procedure [37].

Thus, the proposed approach [37] allows us to solve severalproblems at once:(1) to obtain the Einstein (Debye) characteristic temperatures esti-

mated otherwise from the heat capacity or somehow else;(2) to describe the temperature dependence of the thermal atomic

vibrations using an appropriate analytical function;

Figure 3.7 Temperature dependencies of ueq in the crystals of LuNB12(N = 10, 11, nat). The Einstein (a) and Debye (b) models are used respec-tively for Lu and B atoms. The fit is based on the uobs values marked withsquares [38].



(3) to separate the contributions of the static and dynamic com-ponents into the equivalent parameter ueq of atomic displace-ments.

The Debye and Einstein models were previously used to fit themulti-temperature ADPs and to estimate the Debye (Einstein)temperatures in crystals of various compositions including hex-aborides RB6 (R = Y, La – Gd) [39–42]. This approach, being firstapplied to the dodecaborides LuNB12 (N = 10, 11, nat), revealeda difference in the static components ushift depending on the iso-tope composition of boron [38]. The abbreviation ’nat’ is hereinafterused to refer to natural boron with the ratio 10B :11B≈ 19.8 : 80.2. Re-fined values of TE (TD) and ushift were substituted in the Eqs. (3.3)and (3.4) to draw the curves for Lu and B presented in Figs. 3.7 (a)and (b), respectively.

The ADPs sum up mean-square zero vibrations 〈u2〉zero,temperature-dependent thermal vibrations 〈u2(T)〉, and static

Figure 3.8 Experimental ADPs (ueq) in HoB12 are fitted using the Ein-stein (for Ho) and Debye (for B) models in temperature ranges 86–180 and210–500 K. R = ∑ |u2

obs − u2calc|/ ∑ u2

obs.



shifts 〈u2〉shift. The curves in Fig. 3.7 are plotted for three crys-tals with very close Debye (Einstein) temperatures, so that ueq(T)mostly differ in their temperature independent components〈u2〉c = 〈u2〉zero + 〈u2〉shift. As shown in [38], these componentsare maximal in LunatB12 both for Lu and B atoms. Static distortionsof boron polyhedra are combined with static shifts of Lu atomsfrom the lattice points, which can be explained by disorder in10B -11B substitution in the crystal with natural boron.

The structure of single-crystal HoB12 was studied by x-raydiffraction analysis in the Fm3m group at 29 temperatures inthe range of 86–500 K [43]. Temperature variations of ueq(B) andueq(Ho) lose stability near 200 K. To improve the fit, one has todivide each of the experimental sets of ueq into two parts obtainedin the 86–180 K and 210–500 K temperature ranges, and to buildtwo curves for each atom (Fig. 3.8) for better modeling of the ex-perimental curves. The instability of the unit-cell values could beclearly determined from the x-ray data not only in HoB12 in thetemperature range 150–200 K, but also in RB12 (R = Ho, Tm, Yb,Lu) below 200 K (see Figs. 3.9–3.13 in the next section). The devel-opment of similar lattice instability with decreasing temperaturewas also reported earlier in LuNB12 crystals with different isotopicboron composition (N = 10, 11, nat) that were studied using low-temperature heat capacity and Raman scattering data [24]. Themaximum density of vibrational states was observed at the tem-perature near 150 K [24]. It was noted that the mean free path ofphonons reaches the Ioffe-Regel limit in the vicinity of this temper-ature, being compared with their wavelength. Remarkable spectralchanges in the zero-field spectra and a sharp maximum in therelaxation rate were recorded near 150 K in µSR experiments fordodecaborides RB12 (R = Yb, Lu) and solid solutions Lu1−xYbxB12.It has been suggested that the large-amplitude dynamic featuresarise from atomic motions within the B12 clusters [44, 45]. Mostlikely, the instability of ueq is caused by changes in the phononstructure of the rare-earth dodecaborides and is not a unique fea-ture of only HoB12.



3.4 Crystal structure: problems and results

3.4.1 The Jahn-Teller distortions of structural parameters

Active studies of the RB12 structure at various temperatures,which were started in the early 2000s but not continued at thattime, were resumed later after the tetragonal distortion of theLuB12 structure had been confirmed in the temperature range50–75 K [48]. The structure of a LuB12 single crystal was thenthoroughly studied at room temperature [49]. The single crystalsof LuB12 were grown by modified crucibleless inductive floatingzone melting using high-purity source materials: lutetium oxideLu2O3 and boron [17]. One of the purposes of the re-examinationof the known structure was to assess the suitability of the grownsingle crystals for accurate structure analysis. The refinement ofthe structural model in the Fm3m symmetry group with a uniquelylow residual factor R = 0.2% was made possible due to the highdiffraction quality of the single crystals combined with a set oforiginal experimental techniques [50–52] that ensured the accuracyand reliability of the x-ray data measured.

Besides that, accurate measurements of the periods of theLuB12 crystal lattice were carried out in the temperature range 20–295 K [53]. Two periods a ≈ b did not differ within the limits of thestandard uncertainty (σ), but the third period c steadily deviateddownward by 2σ or more over practically the entire temperaturerange. In absolute values, the difference in the lattice constants isvery small (about 0.002 Å), which is an order of magnitude lessthan in the lattice of ScB12. Such a small difference in the latticeconstants does not give grounds for a revision of the structuralmodel, especially in view of what was said above about the ex-cellent results of the refinement of the cubic structure of LuB12.However, even very small differences in the lattice constants canhave a significant effect on the physical properties of crystals. Thelattice parameters must be determined for many dodecaborides ofdifferent composition in a wide temperature range without sym-metry restrictions in order to collect experimental information onthe Jahn-Teller distortions. This work is still far from complete,both in the number of crystals studied and in the number of tem-


3.4 Crystal structure: problems and results 21

� � � � � � � � � � � � � � �

� � �

� � �

� � � �

� � � �

� � � � � � � � � � � � � � �

� � � �

� � � �

� � � � �

� � � � �

� � � �

�

�

�

��

��

��

��

� � � � � � � � � � � �

�

�

� �

�, �

, ��

��

��

� � � � � � � � � � � �

�

Figure 3.9 Linear (a, b, c) and angular (α, β, γ) unit-cell parameters ofHoB12 in the temperature range 85–500 K [43].

� � � � � � � � �

� � � �

� � � �

� � �

� � �

� � � � � � � � �

� � �

� � � � �

� � � � ��

�

�

��

��

��

��

� � � � � � � � � � � �

�

�

�

�

�, �

, ��

��

��

� � � � � � � � � � � �

�

Figure 3.10 Linear (a, b, c) and angular (α, β, γ) unit-cell parameters ofTmB12 in the temperature range 85–300 K [46].

� � � � � � � � �

� � � �

� � �

� � � �

� � � �

� � � �

� � � � � � � � �

� � � �

� � � � �

� � � � �

�

�

�

��

��

��

��

� � � � � � � � � � � �

�

�

�

�

�, �

, ��

��

��

� � � � � � � � � � � �

�

Figure 3.11 Linear (a, b, c) and angular (α, β, γ) unit-cell parameters ofYbB12 in the temperature range 85–300 K [47].



perature points measured. After the first experiments with LuB12,linear and angular unit-cell parameters have been determined atvarious temperatures for HoB12, TmB12, and YbB12 as shown inFigs. 3.9 – 3.11.

The linear parameters always manifest small tetragonal-typedistortions. Note that one lattice constant of LuB12 and TmB12 issmaller than the other two: a ≈ b > c (the unit cell is slightly com-pressed along an edge), whereas one lattice constant of HoB12 isslightly elongated: a > b ≈ c. Both linear and angular parametersof each unit cell undergo the most noticeable non-linear changesin the same low-temperature region between 100 and 150 K. It isnoteworthy that lattices of TmB12 and YbB12 undergo oppositechanges despite the proximity of Tm and Yb in the series of rareearth elements. Closer to the middle of the mentioned temperaturerange, the lattice constants of YbB12 abruptly decrease and returnto the former, even slightly larger values with a further decrease intemperature. The obliquity of the YbB12 lattice slightly increases,but then the angles return to their previous values. At the sametemperatures, the periods of the TmB12 lattice slightly increase,and the angles become slightly closer to 90◦.

3.4.2 Structural peculiarities of dodecaborides different inisotopic boron composition

Since the cooperative JT effect is determined by the dynamicsof light boron atoms, one can suppose that isotope substitutions10B -11B may affect both properties and crystal structure of dode-caborides. Till recently, the research was mainly limited to phys-ical properties [9, 17, 24–26]. Thermal expansion of Lu10B12 andLunatB12 was studied based on the x-ray powder diffraction datain the temperature range 10–290 K [54]. Both samples showednegative thermal expansion between 50 and 100 K (Fig. 3.12).

This is consistent with the temperature region in which thenegative thermal expansion was previously observed for LunatB12by the three-terminal capacitive method [15]. The lattice constantof Lu10B12 is increased relative to that of LunatB12 by 0.001–0.002 Åover the measured temperature range. The β-rhombohedral boronlattices have the same property, but the difference between the



Figure 3.12 The lattice parameter vs. temperature in LuB12 containingnatural boron and 10B isotope [8].

lattice parameters in the crystals with 10% and 97% content of 10Bis more noticeable (about 0.03 Å) as established in Ref. [55] whoseauthors presented a theoretical justification for such an expansionof the 10B lattice.

An influence of the isotopic composition on the structure andproperties of LuNB12, N = 10, 11, nat, was studied in [38]. Takinginto account both linear and angular distortions of the unit cellsof the three crystals, one can conclude that the Lu10B12 lattice isdistorted rather by tetragonal type a ≈ b > c whereas the dis-tortions of the Lu11B12 lattice are more similar to pseudo-trigonalones a ≈ b ≈ c, α ≈ β ≈ γ > 90◦. Lattice distortions of LunatB12have an intermediate character (see Fig. 3.13).

At temperatures below 140 K, the distortions are nonlinear, ascan be assumed despite the small number of the points measured.Nonlinear distortions of the parameters, which occur at close tem-peratures in three different crystals, are hardly explained by a sheeraccident. At a temperature of about 120 K, the trigonal-type distor-tions of the lattices of Lu11B12 and LunatB12 are amplified, as wellas the pseudo-tetragonal lattice distortions of Lu10B12 crystal, butthe situation changes again with a further decrease in temperature.Thus, we observe the same jump in the parameters of the unit cell



� � � � � � � � �

� � � �

� � � �

� � �

� � � �

� � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � �

� � � � �

� � � � �

� � � � �

� � � � �

� � � � � � � � � � � � � � � � � �

��

��

�

��

��

��

��

� � � � � � � � � �

� ��

��

��

� �

��

�

� � � � � � � � � �

��

��

��

� ��

��

� � � � � � � � � �

��

��

��

� � �

� � �

�

�

��

��

��

��

�

� � � � � � � � � �

��

��

��

�

�

� � � � � � � � � �

��

��

��

�

� � � � � � � � � �

��

��

��

Figure 3.13 Temperature dependences of the lattice parameters forLuNB12 over the temperature range 88–293 K: (a) lattice constants; (b) unit-cell angles. Experimental values are connected by dashed lines; solid linesin panel (a) are linear fits. Standard uncertainties do not exceed 0.0002 Åand 0.001◦, respectively [38].

approximately in the middle of the temperature range 100–150 Kas in other three dodecaborides mentioned above. It is worth not-ing that 120 K is close to the upper boundary of the temperatureinterval with negative thermal expansion of LunatB12 according toRefs. [14, 15].

3.4.3 Formation of charge stripes in voids of the crystallattice

The numerical differences between the lattice parameters are verysmall and do not require a transition to the low-symmetry structuremodel. The crystal structures of LuB12, TmB12, HoB12 and manyother dodecaborides can be successfully refined in the cubic groupFm3m with low values of R-factors. It should be noted, however,



that the completeness of the structural analysis is judged not onlyby the R-factor value but also by the distribution of the residualelectron density (ED) on the difference Fourier maps. Fourier syn-thesis of the electron density is a computational procedure, whichstarts with a set of both experimental and previously calculatedparameters. The computational formula can be written in generalterms as follows:

G(r) =1V ∑

HA(H) exp[iϕ(H)] exp(−2πi H·r). (3.5)

Here G(r) is either full (g) or residual (∆g) electron density, re-sulting respectively either from a “regular” or difference Fouriersynthesis; V is the unit-cell volume; H = ∑i hia

∗i is a scattering

vector; and ϕ(H) is a scattering phase. A(H) are coefficients de-pendent on the type of the Fourier synthesis. In case of differenceFourier synthesis, A(H) =

∣∣|Fobs(H)| − |Fcalc(H)|∣∣ is a difference

between observed and calculated absolute values of the structurefactor. The first value is the square root of the reflection intensitywhereas the second one is calculated from atomic coordinates andADPs, whose values are refined using a least-square technique.

As follows from Eq. (3.5), the Fourier synthesis of the electrondensity does not require any data on the crystal symmetry. It canbe performed independently in each point of the crystal lattice.Nevertheless, the symmetry of the crystal is usually taken intoconsideration in the algorithms that implement Fourier synthesisof the electron density. It means that the measured intensities ofx-ray reflections are averaged in the corresponding Laue class andthe Fourier synthesis is performed in a symmetrically independentregion of the unit cell. As a result, the symmetry of the Fourier mapexactly corresponds to the space group, information about whichis fed to the input of the computational procedure. Certainly, anymeasurement is not free from the influence of instrumental errorsand the data processing methods. On the one hand, the above-mentioned techniques of calculations are designed to improvethe accuracy of the results and to ensure visual consistency ofthe Fourier maps with the stated symmetry of the crystals. Onthe other hand, averaging can harm since the symmetry of theelectron-density distribution over the cell can be overestimated.



Figure 3.14 Difference Fourier maps (residual electron density ∆g ine/Å3) in the x = 0 face of the LuB12 unit cell at (a) 90 K and (b) 295 K. Redcircle is the Lu site; green circles are B sites. The panels (c) – (g) and (d) – (e)are the surface plots of difference Fourier maps in the vicinity of the Luion, in the x = 0, y = 0, and z = 0 faces of the unit cell, respectively. Thefirst and second rows of the figure correspond to temperatures 90 K and295 K, respectively [27].

In the case when accuracy and reliability of measured x-ray dataare ensured by reliable measurement of literally each reflection,with subsequent consideration of experimental corrections usingspecial techniques [50–52], one may feed a less symmetrical groupto the input of the Fourier procedure. In [27], this approach wasapplied to LuB12 whose structure was first refined in the high-symmetry Fm3m group at temperatures 295 and 90 K. After that,the measured values of |Fobs|were averaged in the mmm Laue classinstead of m3m, and the orthorhombic Fmmm group was fed to theinput of the difference Fourier procedure, skipping the tetragonalI4/mmm group, which would require a transition to another unitcell. The difference Fourier maps built from low-temperature (90 K)x-ray data clearly showed residual electron-density peaks orientedalong [001] at distances of about 0.5 Å from the central position ofLu. As can be seen from Fig. 3.14, similar peaks are absent along[010], which is symmetrically equivalent to [001] in the cubic group.This result agrees with the result obtained in the same work [27]



Figure 3.15 Residual electron-density distribution in the x = 0, y = 0,and z = 0 planes of LuB12. Difference Fourier synthesis is done in F1using data collected at four temperatures. Contour intervals are 0.2 e/Å3

(295, 135, 95 K) and 1 e/Å3 (50 K). Positive (pink) and negative (light-green) residual electron density is highlighted. The central Lu(0, 0, 0) site(lime green circle) is surrounded by eight boron sites (dark green circles);[−0.5, 0.5] intervals are periods of the crystal lattice [33].



Figure 3.16 Maximum-entropy-method maps are calculated from theLuB12 data sets collected at temperatures 293, 135, 95 and 50 K. Threecolumns from left to right present thin slices of the electron-density distri-bution in three planes of the crystal lattice. The central Lu is surroundedby eight boron atoms; [−0.5, 0.5] intervals are periods of the crystal lat-tice [33].



concerning unequal magnetoresistance in the LuB12 sample in twodirections of the 〈100〉 family.

In the next work [33], the crystal structure of LuB12 was studiedat the four temperatures 293, 135, 95 and 50 K. To eliminate pos-sible dependence of the results on systematic instrumental errorsand on the features of the crystalline sample, the x-ray experimentswere performed on three different-type diffractometers and on twoLuB12 crystals. To analyze the electron-density distribution in thecrystal at room temperature, the same data were used that werepreviously collected on a CAD4 diffractometer (Enraf Nonius) fora precise analysis of the cubic structure of LuB12 [49]. The x-raydata at 135 and 95 K were collected on an Xcalibur EOS S2 diffrac-tometer with a two-dimensional CCD detector. The experimentat 50 K was obtained on a four-circle Huber-5042 diffractometerequipped with a point detector and a closed-cycle helium cryostatDisplex DE-202. The structure was first refined in Fm3m as before,but information on triclinic F1 symmetry was fed to the input ofthe Fourier procedure. Non-standard abbreviation F1 instead of P1is due to the reluctance to move to another (non-cubic) cell, whichwould correspond to the standard setting. The difference Fouriermaps were built for each temperature in three sections of a crystalwith the (100), (010), (001) planes. As seen from Fig. 3.15, the sym-metry of the residual electron-density distribution is clearly lowerthan orthorhombic. The selected directions remain but lose theirexact orientation along the canceled axis 2 of the orthorhombicgroup, turning in the direction closer to the face diagonal of theunit cell. The residual electron density increases almost by an or-der of magnitude at the temperature of 50 K forming a continuousdiagonal strip in the (010) section. The formation of the electron-density strip at 50 K is confirmed by the maximum entropy method(MEM) as shown in Fig. 3.16.

We associate this observation with the formation of a filamen-tary structure of conductive channels — charge stripes along se-lected directions in the crystal [33]. In the same paper, two resultswere compared, which were obtained on LuB12 samples cut fromone block. The same sample could not be used in all experimentsdue to different requirements for its size and shape for x-ray exper-iments and measurements of transport and magnetic properties.



Moreover, the x-ray measurements were carried out at significantlyhigher temperatures and in the absence of an external magneticfield. The more surprising is the exact orientational coincidence oftwo pictures in the left and right parts of Fig. 3.17, one of which(left) illustrates the anisotropy of the transverse magnetoresistancein LuB12 whereas the second picture demonstrates the anisotropyof the residual electron-density distribution in LuB12 at 50 K.

Another structure of a single-crystal Tm0.19Yb0.81B12 was an-alyzed according to the same scheme at room temperature [56].Extreme members TmB12 and YbB12 in a series of solid solutionsTm1−xYbxB12 vary greatly in their properties, despite the proxim-ity of Tm and Yb in the series of rare-earth elements. Unlike metal-lic TmB12 with antiferromagnetic properties, YbB12 is a narrow-gapsemiconductor known as a Kondo insulator. In order to analyze theloss of metallic properties when thulium is replaced by ytterbium,the information is needed on the corresponding changes in thecrystal structure.

It has been determined that the crystal lattice of Tm0.19Yb0.81B12has the same type of distortion as that of LuB12, with a ≈ b >c and a small difference of about 0.002 Å between the smallerlattice constant and the other two. The residual electron densityis oriented predominantly along the three face diagonals of the

Figure 3.17 (a) Magnetoresistance anisotropy of LuB12 in polar coordi-nates: ∆ρ/ρ0 = [ρ(ϕ, B)− ρ(ϕ0, B)]/ρ(ϕ0, B), ϕ0 = 270◦ correspondingto B ‖ [110]; (b) anisotropic electron-density distribution in a thin layer ofthe electron density reconstructed by the maximum entropy method [33].


3.5 Conclusions 31

unit cell. They are connected by a spatial diagonal, which is one ofthe three-fold axes of the undistorted cubic structure. The residualelectron density forms a strip along one of the face diagonals evenat room temperature, as can be seen in Fig. 3.18.

3.5 Conclusions

The results presented in this chapter demonstrate the complexity ofthe atomic structure of the dodecaborides, a complete descriptionof which does not fit into the framework of a simple cubic model.Both atomic coordinates being expressed in fractions of the latticeconstants and ADPs of almost all dodecaborides correspond well tocubic symmetry and do not require revision of the structural modeldespite the Jahn-Teller distortion of lattice parameters. Symmetryviolations manifest themselves in difference Fourier syntheses as

Figure 3.18 (a) Difference Fourier and (b) maximum-entropy-methodmaps of Tm0.19Yb0.81B12 are created in (100), (010), (001) faces of the unitcell. Electron density (g) in the layer of any given thickness is automaticallydivided into several levels from gmin to gmax, each of them is assigned toa definite color from dark-blue over green to red. The values of gMEM arecut at the level gmax = 0.075% of the maximal gMEM value to show fineelectron-density gradations in the thin layer. Difference electron-densityvalues are cut at ±0.5 e/Å3 [56].



an asymmetric distribution of the residual electron density in theinterstices of the crystal lattice along symmetrically equivalentdirections. These results are in good agreement with the observedasymmetry of physical properties (conductive, magnetic).

Another prospective direction of the structural analysis of dode-caborides is the quantitative analysis of the temperature behaviorof the atomic displacement parameters using multi-temperature x-ray data. The dynamics of the crystal lattice can be traced withoutgoing beyond the cubic structural model, by matching the equiva-lent atomic displacement parameters to the extended Einstein orDebye models.

The analysis of the dodecaboride structure is thus not limited tothe refinement of the structural model in the high-symmetric groupat one or several temperatures. The multi-temperature data onADPs must be supplemented with the temperature dependent unit-cell parameters, which are not bound by the symmetry constraints,and with the difference Fourier maps built without reliance on thesymmetry of the structure model.

The transition from single experiments to systematic researchof the structure-property relationship in dodecaborides requiresthe creation of a database of diffraction data. For reliable char-acterization of a single dodecaboride of a certain composition, itis necessary to carry out a series of diffraction experiments in awide temperature range with the maximum possible coverage ofthe low-temperature region. The temperature step should be se-lected individually for each composition in order to monitor thestructural parameters.

Acknowledgments

The authors are grateful to N. E. Sluchanko and N. Yu. Shitsevalovafor useful discussions. This work was supported by the Ministryof Science and Higher Education within the state assignment ofthe Federal Scientific Research Center (FSRC) “Crystallographyand Photonics” of the Russian Academy of Sciences in the partrelated to the development of structural analysis methods. Crys-tal structures and properties of HoB12 and ErB12 crystals were


References 33

studied with the support of the Russian Foundation for Basic Re-search, Grant No. 18-29-12005; similar studies on TmB12, YbB12,and LuB12 were supported by the Russian Science Foundation,grant No. 17-12-01426. The diffraction data were collected usingthe equipment of the Shared Research Center of the FSRC “Crystal-lography and Photonics” of the Russian Academy of Sciences andwas supported by the Russian Ministry of Education and Science(project RFMEFI62119X0035).

References

[1] Bertaut, F. and Blum, P.; “La structure des borures d’uranium”;Comptes Rendus Acad. Sci. 229, 666–667 (1949).

[2] Post, B. and Glaser, F. W.; “Crystal structure of ZrB12”; J. Miner. Met.Mater. Soc. 4, 631–632 (1952).

[3] La Placa, S., Binder, I., and Post, B.; “Binary dodecaborides”; J. Inorg.Nuclear Chem. 18, 113–117 (1961).

[4] Przybylska, M., Reddoch, A. H., and Ritter, G. J.; “The preparationand structure of lutetium diboride, scandium dodecaboride andlutetium antimonide”; J. Am. Chem. Soc. 85, 407–411 (1963).

[5] Matkovich, V. I., Economy, J., Giese, R. F., and Barrett, R.; “Thestructure of metallic dodecarborides”; Acta Crystallogr. 19, 1056–1058(1965).

[6] Matkovich, V. I. and Economy, J.; “Structural determinants in thehigher borides”; in Matkovich, V. I., Samsonov, G. V., Hagenmuller,P., and Lundstrom, T. (eds.), Boron and refractory borides, pp. 78–95,(Springer, Berlin, Heidelberg, 1977).

[7] Flachbart, K., Alekseev, P., Grechnev, G., Shitsevalova, N.,Siemensmeyer, K., Sluchanko, N., and Zogal, O.; “Rare-earth do-decaborides — magnetism, superconductivity and other properties”;chap. 2 in Delfrey, K. N. (ed.), Rare Earths: Research and Applications,vol. 34, pp. 79–125, (Nova Science Publishers, USA, 2008).

[8] Mori, T.; “Higher borides”; chap. 238 in Gschneidner Jr., K. A., Bünzli,J.-C. G., Pecharsky, V. K. (eds.), Handbook on the Physics and Chemistryof Rare Earths, vol. 38, pp. 105–173, (North-Holland2008).

[9] Werheit, H., Filipov, V., Shirai, K., Dekura, H., Shitsevalova, N.,Schwarz, U., and Armbrüster, M.; “Raman scattering and isotopicphonon effects in dodecaborides”; J. Phys.: Condens. Matter 23, 065403(2011).

[10] Cannon, J. F., Cannon, D. M., and Tracy Hall, H.; “High pressuresyntheses of SmB2 and GdB12”; J. Less Common Metals 56, 83–90(1977).



[11] Akopov, G.; “Rediscovering the crystal chemistry of superhard do-decaborides and other higher borides”; Ph.D. thesis; University ofCalifornia, Los Angeles (2018).

[12] Liang, Y., Zhang, Y., Jiang, H., Wu, L., Zhang, W., Heckenberger, K.,Hofmann, K., Reitz, A., Stober, F. C., and Albert, B.; “Thermodynamicground states of multifunctional metal dodecaborides”; Chem. Mat.31, 1075–1083 (2019).

[13] Leithe-Jasper, A., Sato, A., and Tanaka, T.; “Refinement of the crystalstructure of zirconium dodecaboride, ZrB12, at 140 K and 293 K”;Z. Kristallogr. NCS 217, 319–320 (2002).

[14] Czopnik, A., Shitsevalova, N., Krivchikov, A., Pluzhnikov, V.,Paderno, Y., and Onuki, Y.; “Thermal properties of rare earth do-decaborides”; J. Solid State Chem. 177, 507–514 (2004).

[15] Czopnik, A., Shitsevalova, N., Pluzhnikov, V., Krivchikov, A.,Paderno, Y., and Onuki, Y.; “Low-temperature thermal properties ofyttrium and lutetium dodecaborides”; J. Phys.: Condens. Matter 17,5971 (2005).

[16] Menushenkov, A. P., Yaroslavtsev, A. A., Zaluzhnyy, I. A., Kuznetsov,A. V., Chernikov, R. V., Shitsevalova, N. Y., and Filippov, V. B.; “Fea-tures of the local structure of rare-earth dodecaborides RB12 (R = Ho,Er, Tm, Yb, Lu)”; JETP Lett. 98, 165–169 (2013).

[17] Werheit, H., Paderno, Y., Filippov, V., Paderno, V., Pietraszko, A.,Armbrüster, M., and Schwarz, U.; “Peculiarities in the Raman spectraof ZrB12 and LuB12 single crystals”; J. Solid State Chem. 179, 2761–2767(2006).

[18] Pietraszko, A., Czopnik, A., Shitsevalova, N., Paderno, Y., and Pluzh-nikov, V.; “Crystal structure and thermal expansion of LuB12 singlecrystals”; Acta Crystallogr. Sect. A 56 (Supplement), s421 (2000).

[19] Hamada, K., Wakata, M., Sugii, N., Matsuura, K., Kubo, K., andYamauchi, H.; “Phase transition in the Zr1−xScxB12 system”; Phys.Rev. B 48, 6892–6898 (1993).

[20] Paderno, Y. and Shitsevalova, N.; “Stabilization of cubic scandiumdodecaboride”; J. Alloy. Compd. 219, 119–123 (1995).

[21] Kohout, A., Batko, I., Czopnik, A., Flachbart, K., Matas, S., Meissner,M., Paderno, Y., Shitsevalova, N., and Siemensmeyer, K.; “Phase dia-gram and magnetic structure investigation of the fcc antiferromagnetHoB12”; Phys. Rev. B 70, 224416 (2004).

[22] Siemensmeyer, K., Flachbart, K., Gabáni, S., Mat’aš, S., Paderno,Y., and Shitsevalova, N.; “Magnetic structure of rare-earth dodeca-borides”; J. Solid State Chem. 179, 2748–2750 (2006).

[23] Siemensmeyer, K., Habicht, K., Lonkai, T., Mat’aš, S., Gabáni, S.,Shitsevalova, N., Wulf, E., and Flachbart, K.; “Magnetic propertiesof the frustrated fcc–antiferromagnet HoB12 above and below TN”;J. Low Temp. Phys. 146, 581–605 (2007).


References 35

[24] Sluchanko, N. E., Azarevich, A. N., Bogach, A. V., Vlasov, I. I.,Glushkov, V. V., Demishev, S. V., Maksimov, A. A., Tartakovskii,I. I., Filatov, E. V., Flachbart, K., Gabáni, S., Filippov, V. B., Shitseval-ova, N. Y., and Moshchalkov, V. V.; “Effects of disorder and isotopicsubstitution in the specific heat and Raman scattering in LuB12”;J. Exp. Theor. Phys. 113, 468–482 (2011).

[25] Sluchanko, N. E., Azarevich, A. N., Bogach, A. V., Gavrilkin, S. Y.,Glushkov, V. V., Demishev, S. V., Dukhnenko, A. V., Lyashchenko,A. B., Mitsen, K. V., and Filipov, V. B.; “10B -11B isotope substitutionand superconductivity in ZrB12”; JETP Lett. 94, 642–646 (2011).

[26] Sluchanko, N. E., Azarevich, A. N., Bogach, A. V., Glushkov, V. V.,Demishev, S. V., Anisimov, M. A., Levchenko, A. V., Filipov, V. B., andShitsevalova, N. Y.; “Hall and transverse even effects in the vicinityof a quantum critical point in Tm1−xYbxB12”; J. Exp. Theor. Phys. 115,509–526 (2012).

[27] Sluchanko, N., Bogach, A., Bolotina, N., Glushkov, V., Demishev, S.,Dudka, A., Krasnorussky, V., Khrykina, O., Krasikov, K., Mironov, V.,Filipov, V. B., and Shitsevalova, N.; “Rattling mode and symmetrylowering resulting from the instability of the B12 molecule in LuB12”;Phys. Rev. B 97, 035150 (2018).

[28] Franz, R. and Werheit, H.; “Jahn-Teller effect of the B12 icosahedronand its general influence on the valence band structures of boron-richsolids”; Europhys. Lett. 9, 145–50 (1989).

[29] Franz, R. and Werheit, H.; “Influence of the Jahn-Teller effect onthe electronic band structure of boron-rich solids containing B12icosahedra”; AIP Conf. Proc. 231, 29–36 (1991).

[30] Bersuker, I. B. and Polinger, V. Z. (eds.); “Vibronic interactions inmolecules and crystals”; Springer Series in Chemical Physics vol. 49,(Springer, Berlin, Heidelberg, 1989).

[31] Gehring, G. A. and Gehring, K. A.; “Co-operative Jahn-Teller effects”;Rep. Prog. Phys. 38, 1–89 (1975).

[32] Kaplan, M. D. and Vekhter, B. G.; “Cooperative phenomena in Jahn-Teller crystals”, (Plenum Press, New York, 1995).

[33] Bolotina, N. B., Dudka, A. P., Khrykina, O. N., Krasnorussky, V. N.,Shitsevalova, N. Y., Filipov, V. B., and Sluchanko, N. E.; “The lowersymmetry electron-density distribution and the charge transportanisotropy in cubic dodecaboride LuB12”; J. Phys.: Condens. Matter30, 265402 (2018).

[34] Trueblood, K. N., Bürgi, H.-B., Burzlaff, H., Dunitz, J. D., Gramaccioli,C. M., Schulz, H. H., Shmueli, U., and Abrahams, S. C.; “Atomicdisplacement parameter nomenclature. Report of a subcommitteeon atomic displacement parameter nomenclature.”; Acta Crystallogr.Sect. A 52, 770–781 (1996).



[35] Einstein, A.; “Die Plancksche Theorie der Strahlung und die Theorieder spezifischen Wärme”; Ann. Phys. 327, 180–190 (1907).

[36] Debye, P.; “Vorträge über die kinetische Theorie der Materie und derElektrizität”; Mathematische Vorlesungen an der Universität Göttingen,vol. 6, (B.G. Teubner, Leipzig, 1914).

[37] Dudka, A. P., Bolotina, N. B., and Khrykina, O. N.; “DebyeFit: asimple tool to obtain an appropriate model of atomic vibrations insolids from atomic displacement parameters obtained at differenttemperatures”; J. Appl. Crystallogr. 52, 690–692 (2019).

[38] Bolotina, N. B., Dudka, A. P., Khrykina, O. N., Glushkov, V. V., Azare-vich, A. N., Krasnorussky, V. N., Gabáni, S., Shitsevalova, N. Y.,Dukhnenko, A. V., Filipov, V. B., and Sluchanko, N. E.; “On the roleof isotopic composition in crystal structure, thermal and charge-transport characteristics of dodecaborides LuNB12 with the Jahn-Teller instability”; J. Phys. Chem. Solids 129, 434–441 (2019).

[39] Korsukova, M.; “Vacancies and thermal vibrations of atoms in thecrystal structure of rare earth hexaborides”; Jpn. J. Appl. Phys. Se-ries 10, 15–18 (1994); Proc. 11th Int. Symp. Boron, Borides and Rel.Compd., Tsukuba (Japan), 1993.

[40] Takahashi, Y., Ohshima, K., Okamura, F. P., Otani, S., and Tanaka, T.;“Crystallographic parameters of atoms in the single crystals of thecompounds RB6 (R = Y, La, Ce, Nd, Sm, Eu, Gd)”; J. Phys. Soc. Jpn.68, 2304–2309 (1999).

[41] Trounov, V. A., Malyshev, A. L., Chernyshov, D. Y., Korsukova, M. M.,Gurin, V. N., Aslanov, L. A., and Chernyshev, V. V.; “Temperaturedependences of the parameters of atoms in the crystal structure of theintermediate-valence semiconductor SmB6: investigation by high-resolution powder neutron diffraction”; J. Phys.: Condens. Matter 5,2479–2488 (1993).

[42] Trounov, V. A., Malyshev, A. L., Chernyshov, D. Y., Korsukova, M. M.,and Gurin, V. N.; “Thermal vibrations and static displacements ofatoms in the crystal structure of neodymium and samarium hexa-borides”; Fiz. Tverd. Tela 36, 2687–2694 (1994).

[43] Khrykina, O. N., Dudka, A. P., Bolotina, N. B., Sluchanko, N. E.,and Shitsevalova, N. Y.; “Structural instability and poorly definedphase transitions in rare-earth dodecaborides RB12 (R = Ho–Lu) atintermediate temperatures”; Solid State Sci. 107, 106273 (2020).

[44] Kalvius, G. M., Noakes, D. R., Wäppling, R., Kratzer, A., Schreier, E.,Iga, F., Takabatake, T., and Löhneysen, H.; “µSR spectroscopy of theKondo insulators Lu1−xYbxB12”; Physica B: Condens. Matter 312–313,210–212 (2002); the International Conference on Strongly CorrelatedElectron Systems.

[45] Kalvius, G. M., Noakes, D. R., Marcano, N., Wäppling, R., Iga, F., andTakabatake, T.; “Dynamics of the internal field in RB12 (R = Er, Yb,


References 37

Lu)”; Physica B: Condens. Matter 326, 398–402 (2003).[46] Dudka, A. P., Khrykina, O. N., Bolotina, N. B., and Shitsevalova, N. Y.;

“Jahn-Teller lattice distortions and asymmetric electron density dis-tribution in the structure of TmB12 dodecaboride in the temperaturerange of 85–293 K”; Crystallogr. Rep. 64, 737–742 (2019).

[47] Dudka, A. P., Khrykina, O. N., Bolotina, N. B., Azarevich, A. N.,Gavrilkin, S. Y., and Sluchanko, N. E.; “Fine details of the crys-tal structure and atomic vibrations in YbB12 with a metal-insulatorphase transition”; in preparation.

[48] Bolotina, N. B., Verin, I. A., Shitsevalova, N. Y., Filippov, V. B., andSluchanko, N. E.; “Structural features of single crystals of LuB12upon a transition to the cage-glass phase”; Crystallogr. Rep. 61, 181–186 (2016).

[49] Dudka, A. P., Khrykina, O. N., Bolotina, N. B., Shitsevalova, N. Y., Fil-ipov, V. B., and Sluchanko, N. E.; “An exceptionally-high diffractionquality dodecaboride LuB12: Growth and single-crystal structure”;J. Alloy. Compd. 692, 535–544 (2017).

[50] Dudka, A.; “ASTRA–a program package for accurate structure anal-ysis by the intermeasurement minimization method”; J. Appl. Crys-tallogr. 40, 602–608 (2007).

[51] Dudka, A.; “New approaches to scaling data measured on a CCDdiffractometer”; J. Appl. Crystallogr. 43, 1440–1451 (2010).

[52] Dudka, A. P.; “ASTRA 4.0 Program: Data reduction for obtainingstructure results of extreme accuracy”; Crystallogr. Rep. 63, 1051–1056(2018).

[53] Dudka, A. P., Smirnova, E. S., Verin, I. A., and Bolotina, N. B.; “Algo-rithm and program for precise determination of unit-cell parametersof single crystal taking into account the sample eccentricity”; Crystal-logr. Rep. 62, 651–659 (2017).

[54] Mori, T., Gumeniuk, R., Grin, Y., Vasylechko, L., Dementiy, G., andShitsevalova, N.; “Thermal expansion and boron isotope effect ofLuB12”; HASYLAB Annual Report, pp. 873–874 (2007).

[55] Gabunia, D., Tsagareishvili, O., Chkhartishvili, L., and Gabunia, L.;“Isotopic composition dependences of lattice constant and thermalexpansion of β-rhombohedral boron”; J. Phys. Conf. Ser 176, 012022(2009).

[56] Sluchanko, N. E., Azarevich, A. N., Bogach, A. V., Bolotina, N. B.,Glushkov, V. V., Demishev, S. V., Dudka, A. P., Khrykina, O. N., Fil-ipov, V. B., Shitsevalova, N. Y., Komandin, G. A., Muratov, A. V.,Aleshchenko, Y. A., Zhukova, E. S., and Gorshunov, B. P.; “Obser-vation of dynamic charge stripes in Tm0.19Yb0.81B12 at the metal-insulator transition”; J. Phys.: Condens. Matter 31, 065604 (2019).

Documents

Chapter 3 arXiv:2010.16239v1 [cond-mat.mtrl-sci] 29 Oct 2020 · for the boron atoms to describe the dynamics of the crystal lattice. 3.1 Introduction There is a wide variety of borides